Estimating a conductivity distribution via a FEM based nonlinear bayesian method
15th of January 1998
Eur. Phys. J. AP 1, 87-91 (1998)
Electrical Impedance Tomography (EIT) of closed conductive media is an ill-posed inverse problem. In order to solve the corresponding direct problem, the Finite Elements Method (FEM) provides good accuracy and preserves the non linear dependence of the observation set upon the conductivity distribution. In this paper, we show that the Bayesian approach presented in  for linear inverse imaging problems is also valid for a non linear problem such as EIT. Our contribution is based on an edge-preserving Markov model as prior for conductivity distribution. Maximum a posteriori reconstruction results from 40 dB noisy measurements (simulated with a finer mesh) yield significant resolution improvement compared to classical methods.
PACS: 02.60.Lj – Ordinary and partial differential equations; boundary value problems / 02.60.Pn – Numerical optimization / 02.70.Dh – Finite-element and Galberkin methods
Reconstruction bayésienne en tomographie d'impédance électrique
Colloque Numélec Lyon, mars 1997
La tomographie d'impédance électrique (TIÉ) des milieux conducteurs fermés est un problème inverse mal posé.
La méthode des éléments finis (MÉF), utilisée pour résoudre le problème direct, préserve la non-linéarité des observations vis-à-vis de la conductivité. Pour la reconstruction, nous introduisons un modèle de douceur a priori, markovien, sur la log-conductivité recherchée, permettant toutefois la restauration de discontinuités. L'estimateur du maximum a posteriori (MAP), obtenu par minimisation de critère pénalisé, apporte une amélioration notable de la résolution.
Stability and accuracy of ﬁnite element direct solvers for Electrical Impedance Tomography
Laboratoire des Signaux et Systèmes, Research Report
Electrical impedance tomography (EIT) of closed conductive media is an ill-posed inverse problem. In the general case, the resolution of the direct problem, derived from a second order partial derivative equation, requires a numerical approximation. The solution brought by the Finite Element Method (FEM) is often used in EIT, because it preserves the nonlinear dependence of the observation set upon the conductivity distribution.
This paper addresses the reliability of numerical FEM direct solvers, as basic tools in 2D EIT inversion methods. Reliability is closely related to the discretization error of the FEM model but also to its numerical stability. Finely discretized FEM models yield reduced discretization errors. Meanwhile, special attention must be paid to stability since a more accurate FEM direct model involves a magniﬁcation of input round-oﬀ errors. A theorem is established that allows easy checking of the numerical stability through the computation of the condition number of the uniform stiﬀness matrix of the FEM solver. When the latter is stable, global inaccuracy reduces to the discretization error of the FEM approximation. Simulations reveal that, provided input current patterns are spatially smooth, the variations of the conductivity distribution have a limited but non negligible inﬂuence on the discretization error.